词条 | Mohr–Mascheroni theorem |
释义 |
In mathematics, the Mohr–Mascheroni theorem states that any geometric construction that can be performed by a compass and straightedge can be performed by a compass alone. It must be understood that by "any geometric construction", we are referring to figures that contain no straight lines, as it is clearly impossible to draw a straight line without a straightedge. It is understood that a line is determined provided that two distinct points on that line are given or constructed, even though no visual representation of the line will be present. The theorem can be stated more precisely as:[1] Any Euclidean construction, insofar as the given and required elements are points, may be completed with the compass alone if it can be completed with both the compass and the straightedge together. Though the use of a straightedge can make a construction significantly easier, the theorem shows that any set of points that fully defines a constructed figure can be determined with compass alone, and the only reason to use a straightedge is for the aesthetics of seeing straight lines, which for the purposes of construction is functionally unnecessary. HistoryThe result was originally published by Georg Mohr in 1672,[2] but his proof languished in obscurity until 1928.[3][4][5] The theorem was independently discovered by Lorenzo Mascheroni in 1797 and it was known as Mascheroni's Theorem until Mohr's work was rediscovered.[6] Motivated by Mascheroni's result, in 1822 Jean Victor Poncelet conjectured a variation on the same theme. He proposed that any construction possible by straightedge and compass could be done with straightedge alone. The one stipulation though is that a single circle with its center identified must be provided. The Poncelet-Steiner theorem was proved by Jakob Steiner eleven years later. Constructive proof approachTo prove the theorem, each of the basic constructions of compass and straightedge need to be proven to be possible by using a compass alone, as these are the foundations of, or elementary steps for, all other constructions. These are:
It is understood that a straight line cannot be drawn without a straightedge. A line is considered to be given by any two points, as any two points define a line uniquely, and a unique line can be defined by any two points on it. #2 - A circle through one point with defined centerThis can be done with compass alone quite naturally. There is nothing to prove. #5 - Intersection of two circlesThis construction can be done directly with a compass provided the centers and radii of the two circles are known. Due to the compass-only construction of the center of a circle (given below), it can always be assumed that any circle is described by its center and radius. Indeed, some authors include this in their descriptions of the basic constructions.[7][8][9] #3, #4 - The other constructionsThus, to prove the theorem, there only compass-only constructions for #3 and #4 need to be given. Alternative proofsSeveral proofs of the result are known. Mascheroni's proof of 1797 was generally based on the idea of using reflection in a line as the major tool. Mohr's solution was different.[3] In 1890, August Adler published a proof using the inversion transformation.[10] An algebraic approach uses the isomorphism between the Euclidean plane and the real coordinate space . This approach can be used to provide a stronger version of the theorem.[11] It also shows the dependence of the theorem on Archimedes' axiom (which cannot be formulated in a first-order language). Constructive proofThe following notation will be used throughout this article. A circle whose center is located at point {{mvar|U}} and that passes through point {{mvar|V}} will be denoted by {{math|U(V)}}. A circle with center {{mvar|U}} and radius specified by a number, {{mvar|r}}, or a line segment {{math|{{overline|AB}}}} will be denoted by {{math|U(r)}} or {{math|U(AB)}}, respectively.[12] In general constructions there are often several variations that will produce the same result. The choices made in such a variant can be made without loss of generality. However, when a construction is being used to prove that something can be done, it is not necessary to describe all these various choices and, for the sake of clarity of exposition, only one variant will be given below. However, many constructions come in different forms depending on whether or not they use circle inversion and these alternatives will be given if possible. Some preliminary constructionsTo prove the above constructions #3 and #4, which are included below, a few necessary intermediary constructions are also explained below since they are used and referenced frequently. These are also compass-only constructions. All constructions below rely on #1,#2,#5, and any other construction that is listed prior to it. Compass equivalence theorem (circle translation){{main|Compass equivalence theorem}}The ability to translate, or copy, a circle to a new center is vital in these proofs and fundamental to establish the veracity of the theorem. The creation of a new circle with the same radius as the first, but centered at different point, is the key feature distinguishing the collapsing compass from the modern, rigid compass. The equivalence of a collapsing compass and a rigid compass was proved by Euclid (Book I Proposition 2 of The Elements) using straightedge and collapsing compass when he, essentially, constructs a copy of a circle with a different center. This equivalence can also be established with compass alone. Reflecting a point across a line
Extending the length of a line segment
This construction can be repeated as often as necessary to find a point {{mvar|Q}} so that the length of line segment {{math|{{overline|AQ}}}} = {{math|n}}⋅ length of line segment {{math|{{overline|AB}}}} for any positive integer {{math|n}}. {{clear}}Inversion in a circle
Point {{mvar|I}} is such that the radius {{mvar|r}} of {{math|B(r)}} is to {{mvar|IB}} as {{mvar|DB}} is to the radius; or {{math|1=IB / r = r / DB}}. In the event that the above construction fails (that is, the red circle and the black circle do not intersect in two points),[15] find a point {{mvar|Q}} on the line {{math|{{overline|BD}}}} so that the length of line segment {{math|{{overline|BQ}}}} is a positive integral multiple, say {{mvar|n}}, of the length of {{math|{{overline|BD}}}} and is greater than {{math|r / 2}} (this is possible by Archimede's axiom). Find {{mvar|Q'}} the inverse of {{mvar|Q}} in circle {{math|B(r)}} as above (the red and black circles must now intersect in two points). The point {{mvar|I}} is now obtained by extending {{math|{{overline|BQ' }}}} so that {{math|{{overline|BI}}}} = {{math|n ⋅ {{overline|BQ' }}}}. {{clear}}Determining the center of a circle through three points
Intersection of two non-parallel lines (construction #3)
Intersection of a line and a circle (construction #4)The compass-only construction of the intersection points of a line and a circle breaks into two cases depending upon whether the center of the circle is or is not collinear with the line. Circle center is not collinear with the lineAssume that center of the circle does not lie on the line.
An alternate construction, using circle inversion can also be given.[16]
Circle center is collinear with the line
Thus it has been shown that all of the basic construction one can perform with a straightedge and compass can be done with a compass alone, provided that it is understood that a line cannot be literally drawn but merely defined by two points. See also
Notes1. ^{{harvnb|Eves|1963|loc=p. 201}} 2. ^Georg Mohr, Euclides Danicus (Amsterdam: Jacob van Velsen, 1672). 3. ^1 {{harvnb|Eves|1963|loc=p. 199}} 4. ^Hjelmslev, J. (1928) "Om et af den danske matematiker Georg Mohr udgivet skrift Euclides Danicus, udkommet i Amsterdam i 1672" [Of a memoir Euclides Danicus published by the Danish mathematician Georg Mohr in 1672 in Amsterdam], Matematisk Tidsskrift B , pages 1–7. 5. ^Schogt, J. H. (1938) "Om Georg Mohr's Euclides Danicus," Matematisk Tidsskrift A , pages 34–36. 6. ^Lorenzo Mascheroni, La Geometria del Compasso (Pavia: Pietro Galeazzi, 1797). [https://books.google.com/books?id=HTY4AAAAMAAJ&pg=PR1#v=onepage&q&f=false 1901 edition.] 7. ^{{harvnb|Eves|1963|loc=p. 202}} 8. ^1 {{harvnb|Hungerbühler|1994|loc=p. 784}} 9. ^{{harvnb|Pedoe|1988|loc=p.122}} 10. ^{{harvnb|Eves|1963|loc=p. 198}} 11. ^Arnon Avron, [https://link.springer.com/article/10.1007/BF01222890 "On strict strong constructibility with a compass alone"], Journal of Geometry (1990) 38: 12. 12. ^{{harvnb|Eves|1963|loc=p. 184}} 13. ^{{harvnb|Pedoe|1988|loc=p. 78}} 14. ^{{harvnb|Pedoe|1988|loc=p. 77}} 15. ^{{harvnb|Pedoe|1988|loc=p. 78}} 16. ^1 2 {{harvnb|Pedoe|1988|loc=p. 123}} 17. ^{{harvnb|Eves|1963|loc=p. 199}} 18. ^Pedoe carries out one more inversion at this point, but the points {{mvar|P}} and {{mvar|Q}} are on the circle of inversion and so are invariant under this last unneeded inversion. 19. ^{{harvnb|Eves|1963|loc=p. 200}} References
Further reading
External links
2 : Compass and straightedge constructions|Theorems in geometry |
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